The Michigan Mathematical Journal

On Diamond’s L1 Criterion for Asymptotic Density of Beurling Generalized Integers

Gregory Debruyne and Jasson Vindas

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We give a short proof of the L1 criterion for Beurling generalized integers to have a positive asymptotic density. We in fact prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate m(x)=nkxμ(nk)/nk=o(1) with the Beurling analog μ of the Möbius function.

Article information

Michigan Math. J., Volume 68, Issue 1 (2019), 211-223.

Received: 12 April 2017
Revised: 3 November 2017
First available in Project Euclid: 31 January 2019

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Zentralblatt MATH identifier

Primary: 11N80: Generalized primes and integers
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11M45: Tauberian theorems [See also 40E05] 11N37: Asymptotic results on arithmetic functions


Debruyne, Gregory; Vindas, Jasson. On Diamond’s $L^{1}$ Criterion for Asymptotic Density of Beurling Generalized Integers. Michigan Math. J. 68 (2019), no. 1, 211--223. doi:10.1307/mmj/1548903624.

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